A1043
Title: Minimax optimal rates of convergence in the shuffled and unlinked regression, and deconvolution under vanishing noise
Authors: Debarghya Mukherjee - Boston University (United States) [presenting]
Cecile Durot - Univ. Paris Nanterre (France)
Abstract: Shuffled and unlinked regression represent intriguing challenges that have garnered considerable attention in many fields, including but not limited to ecological regression, multi-target tracking problems, image denoising, etc. However, a notable gap exists in the existing literature, particularly in vanishing noise, i.e., how the estimation rate of the underlying signal scales with the error variance. The aim is to bridge this gap by delving into the monotone function estimation problem under vanishing noise variance, i.e., the error variance is allowed to go to 0 as the number of observations increases. The investigation reveals that, asymptotically, the shuffled regression problem exhibits a comparatively simpler nature than the unlinked regression; if the error variance is smaller than a threshold, then the minimax risk of the shuffled regression is smaller than that of the unlinked regression. On the other hand, the minimax estimation error is of the same order in the two problems if the noise level is larger than that threshold. The analysis is quite general; any smoothness of the underlying monotone link function is not assumed. Because these problems are related to deconvolution, bounds for deconvolution in a similar context are also provided. Through this exploration, the contribution is to understand the intricate relationships between these statistical problems and shed light on their behaviors when subjected to the nuanced constraint of vanishing noise.
